Three-dimensional (3D) Dirac points inheriting relativistic effects from high-energy physics appear as gapless excitations in the topological band theory. Hosting fourfold linear dispersion, they play the central role among various topological phases, such as representing the degeneracy of paired Weyl nodes carrying opposite chiralities. While they have been extensively investigated in solid state systems for electrons, 3D Dirac points have not yet been observed in any classical systems. Here, we experimentally demonstrate 3D photonic Dirac points in the microwave region with an elaborately designed metamaterial, where two symmetrically placed Dirac points are stabilized by electromagnetic duality symmetry. Furthermore, spin-polarized surface arcs (counterparts of Fermi arcs in electronic systems) are demonstrated, which opens the gate toward implementing spin-multiplexed topological surface wave propagation. Closely linked to other exotic states through topological phase transitions, our system offers an effective medium platform for topological photonics.